1. I Sistemi Positivi Realizzazione: esistenza a tempo continuo e minimalità Lorenzo Farina Dipartimento di informatica e sistemistica A. Ruberti Università di Roma La Sapienza, Italy X Scuola Nazionale CIRA di dottorato Antonio Ruberti Bertinoro, 10-12 Luglio 2006

2. 2 The positive realization problem for continuous-time systems Spectrum translation property

3. 3 Existence conditions

4. 4 Examples - I … not to be!

5. 5 Examples - II … not to be!

6. 6 Minimality of Positive Realizations

7. 7 Does positive factorization suffice? For general systems, the minimal inner dimension of a factorization of the Hankel matrix coincides with the minimal order of a realization. Is that true also for positive systems?

8. 8 Does positive factorization suffice? No rotational simmetry, no 3 rd order positive realization...

9. 9 Does positive factorization suffice? No! A positive factorization of the Hankel matrix!

10. 10 A prologue via examples (I)

11. 11 The spectrum must remain unchanged under a rotation of /2 (q+1) radians A prologue via examples (I) (contd.)

12. 12 The spectrum must remain unchanged under a rotation of /4 radians A prologue via examples (I)

13. 13 The Karpelevich theorem

14. 14 The Karpelevich regions n = 3 n = 4

15. 15 hidden pole A prologue via examples (II)

16. 16 Example 3

17. 17 cA x = 0 2 Ab b A bA b 2 @O@O c x = 0 3 cA x = 0 A bA b 3

18. 18 Minimality of Positive Systems NSC for 3 rd order systems

19. 19 {1 (contd.) r2r2 r3r3 Minimality of Positive Systems NSC for 3 rd order systems

20. 20 (contd.) Minimality of Positive Systems NSC for 3 rd order systems

21. 21 (contd.) Minimality of Positive Systems NSC for 3 rd order systems

22. 22 (contd.) Minimality of Positive Systems NSC for 3 rd order systems

23. 23 Minimality for continuous-time positive systems

24. Generation of all positive realizations

25. 25 Example 1